During the following thousand years, the image of the swastika could be found in many cultures around the world, including in China, Japan, India, and southern Europe.
By the Middle Ages, the swastika was a well known, if not commonly used, symbol but was called by many different names:
China - wan
England - fylfot
Germany - Hakenkreuz
Greece - tetraskelion and gammadion
India - swastika
Though it is not known for exactly how long, Native Americans also had long used the symbol of the swastika.
The Original Meaning
The word "swastika" comes from the Sanskrit svastika - "su" meaning "good," "asti" meaning "to be," and "ka" as a suffix.
Until the Nazis used this symbol, the swastika was used by many cultures throughout the past 3,000 years to represent life, sun, power, strength, and good luck.
Even in the early twentieth century, the swastika was still a symbol with positive connotations. For instance, the swastika was a common decoration that often adorned cigarette cases, postcards, coins, and buildings. During World War I, the swastika could even be found on the shoulder patches of the American 45th Division and on the Finnish air force until after World War II.
Change in Meaning
In the 1800s, countries around Germany were growing much larger, forming empires; yet Germany was not a unified country until 1871. To counter the feeling of vulnerability and the stigma of youth, German nationalists in the mid-nineteenth century began to use the swastika, because it had ancient Aryan/Indian origins, to represent a long Germanic/Aryan history.
By the end of the nineteenth century, the swastika could be found on nationalist German volkisch periodicals and was the official emblem of the German Gymnasts' League.
In the beginning of the twentieth century, the swastika was a common symbol of German nationalism and could be found in a multitude of places such as the emblem for the Wandervogel, a German youth movement; on Jörg Lanz von Liebenfels' antisemitic periodical Ostara; on various Freikorps units; and as an emblem of the Thule Society.
Hitler and the Nazis
In 1920, Adolf Hitler decided that the Nazi Party needed its own insignia and flag. For Hitler, the new flag had to be "a symbol of our own struggle" as well as "highly effective as a poster." (Mein Kampf, pg. 495)
On August 7, 1920, at the Salzburg Congress, this flag became the official emblem of the Nazi Party.
In Mein Kampf, Hitler described the Nazis' new flag: "In red we see the social idea of the movement, in white the nationalistic idea, in the swastika the mission of the struggle for the victory of the Aryan man, and, by the same token, the victory of the idea of creative work, which as such always has been and always will be anti-Semitic." (pg. 496-497)
Because of the Nazis' flag, the swastika soon became a symbol of hate, antisemitism, violence, death, and murder.
Hitler (mis-)appropriated a lot of pre-existing symbols for his Aryan mythology garbage. The swastica is simply the most well-known. If you look around at some of the Indian art (from India that is, not the native American) you'll see lots of swasticas.